Problem Bank

Let $f(x) = x$ and $g(x) = x^2 - x + 3$. Suppose there exist $x_1, x_2, \dots, x_n \in \left[0, \dfrac{9}{2}\right]$ such that

Call $S \subseteq A = \{1, \dots, 2045\}$ triple-free if no element of $S$ is three times another. Let $n$ be the number of triple-free subsets of maximum size; write $n = p^aq^b…

Consider the line $l \colon kx - y - k + 1 = 0$ and

The line $l$ meets the grap…

A regular quadrilateral prism $ABCD\text{-}A_1B_1C_1D_1$ has base $ABCD$ a square of side length $3$ and $AA_1=6$. Points $E,F,G$ lie on the segments $A_1D_1$, $AA_1$, $B_1C_1$ res…

The system of inequalities $2m \leqslant x \leqslant 4m$ (in the unknown $x$) has exactly $3$ integer solutions. Find the range of possible values of $m$.

A sequence $\{a_n\}$ satisfies $a_{n+1}=\dfrac 14(a_n-6)^3+6$ ($n\in\mathbb N^{*}$). Determine, with proof, which of the following claims are true.

1. When $a_1=3$, $\{a_n\}$ is d…

The function

has three distinct zeros. Find the range of possible values of the real number $a$.

Let $f(x) = x^2$ for $0 \leqslant x \leqslant 1$, and let $H(a, b)$ be the maximum distance from a point of the graph of $f$ to the line $y = ax + b$. Find the minimum value of $H(…

A sequence $\{a_n\}$ of positive integers satisfies

Find $a_n$.

The parabola $y^2=4x$ has focus $F$. Through $F$, two perpendicular lines meet the parabola at $A,B$ and $C,D$ respectively. Let $M$, $N$ be the midpoints of the chords $AB$, $CD$,…

The function

has maximum plus minimum equal to $8$. Find $m + n$.

Let $\delta=(a_1,a_2,\dots,a_n)$ be a permutation of $\{1,2,\dots,n\}$, and define

Find $\min_{\delta}F(\delt…